Answer:
The maximum 2 is reached when x=2pi,4pi, and 6pi.
The minimum -6 is reached when x=pi, 3pi,and 5pi.
Step-by-step explanation:
So let's first look at cos(x) on interval (0,21].
How many rotations is that? Does it at least contain 1 full rotation? If it contains one full rotation that means all the values from -1 to 1 (inclusive) are tagged? If it doesn't contain a full rotation, we might have to dig a little deeper.
So we know x=0 isn't included and that's when cosine is first 1,but this doesn't mean 1 won't be hit later.
Let's figure out the number of rotations:
21/(2pi)=3.3 approximately
This means we make at least 3 rotations.
So this means we definitely will have all the values from -1 to 1 tagged (inclusive).
Now let's look at whole function.
f(x) = -2 + 4 cos x
-2+(-4) to -2+4 will be the range of the function
So the minimum is -6 and the maximum is 2.
So the min occurs when cos(x)=-1 and the max occurs when cos(x)=1.
We have a little over three rotations and remember we can't include x=0.
cos(x)=1
when x=2pi (one full rotation)
when x=4pi (two full rotations)
when x=6pi (three full rotations)
We will stop here because cosine won't be 1 again until a fourth full rotation
cos(x)=-1
when x=pi (half rotation)
When x=3pi (one + half rotation)
When x=5pi (two+half rotation)
We can't include x=7pi (three+half rotation)
because this one is actually not in the interval because 3.5 is more than 3.3 .
The maximum 2 is reached when x=2pi,4pi, and 6pi.
The minimum -6 is reached when x=pi, 3pi,and 5pi.
